CANeng11 - to answer your question, no. The joist designator is tied to a minimum strength and stiffness requirement. In other words, all SJI member fabricators have to produce a 22K4 that can support the loads listed in the SJI tables at the given spans. How they get there is up to them, and is how they develop their economy and compete with one another.
That said, somebody did put a document together that can be useful for estimating the chord designator for a K series joist. It's attached here. THIS IS NOT A GUARANTEE. These sizes are approximate and is a useful tool for evaluating existing buildings when you don't have drawings, and can also give you some insight into the size material that is LIKELY used.
If you know the designator and the fabricator, tracking down their load tables shouldn't be too bad. From that you can build the typical shear and moment diagrams for a uniformly loaded joist. Compare your new shear and moment diagram and make sure you don't cross it.